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The Lotka–Volterra model makes a number of assumptions, not necessarily realizable in nature, about the environment and evolution of the predator and prey populations:

The Lotka–Volterra model makes a number of assumptions, not necessarily realizable in nature, about the environment and evolution of the predator and prey populations:

* 猎物随时都有充足的食物。

* 猎物随时都有充足的食物。

* 捕食者种群的食物供应完全取决于猎物种群的大小。

* 捕食者种群的食物供应充足与否完全取决于猎物种群的大小。

* 各种群数量变化率与其规模成正比。

* 各种群数量变化率与其规模成正比。

* 在此过程中，环境不会因一种物种而改变，并且无关于遗传适应性。

* 在此过程中，环境不会因一种物种而改变，并且无关于遗传适应性。

The prey are assumed to have an unlimited food supply and to reproduce exponentially, unless subject to predation; this exponential growth is represented in the equation above by the term αx. The rate of predation upon the prey is assumed to be proportional to the rate at which the predators and the prey meet, this is represented above by βxy. If either  or  is zero, then there can be no predation.

The prey are assumed to have an unlimited food supply and to reproduce exponentially, unless subject to predation; this exponential growth is represented in the equation above by the term αx. The rate of predation upon the prey is assumed to be proportional to the rate at which the predators and the prey meet, this is represented above by βxy. If either  or  is zero, then there can be no predation.

假定猎物具有无限的食物供应，除非受到捕食，否则可以成倍繁殖，那么其指数增长由上式中的''αx''来表示。假设掠食者的掠食率，与掠食者和猎物的相遇率成正比，则用上式中的''βxy''表示。注意如果{{mvar|x}}或{{mvar|y}}为零，则表示没有掠夺。

假定猎物具有无限的食物供应，且除非受到捕食，否则可以成倍繁殖，那么其指数增长由上式中的''αx''来表示。假设掠食者的掠食率，与掠食者和猎物的相遇率成正比，则用上式中的''βxy''表示。注意如果{{mvar|x}}或{{mvar|y}}为零，则表示没有捕猎。

In this equation,  represents the growth of the predator population. (Note the similarity to the predation rate; however, a different constant is used, as the rate at which the predator population grows is not necessarily equal to the rate at which it consumes the prey). The term  represents the loss rate of the predators due to either natural death or emigration, it leads to an exponential decay in the absence of prey.

In this equation,  represents the growth of the predator population. (Note the similarity to the predation rate; however, a different constant is used, as the rate at which the predator population grows is not necessarily equal to the rate at which it consumes the prey). The term  represents the loss rate of the predators due to either natural death or emigration, it leads to an exponential decay in the absence of prey.

在此等式中，{{math|''δxy''}}代表捕食者种群的增长。（请注意此处与捕食率表达式虽然相似；但是使用了一个不同的常数，因为捕食者的生长速度不一定等于其消耗猎物的速度）。另外{{math|''γy''}}表示由于自然死亡或迁徙造成的捕食者数量减少率，它在没有猎物的情况下是指数衰减的。

在此等式中，{{math|''δxy''}}代表捕食者种群的增长。（请注意此处与捕食率表达式虽然相似；但是使用了一个不同的常数，因为捕食者的生长速度不一定等于其消耗猎物的速度）。另外{{math|''γy''}}表示由于自然死亡或迁徙造成的捕食者数量减少率，它在没有猎物的情况下是指数型衰减的。

因此，该等式表示，捕食者种群的变化率取决于其捕杀猎物的速率减去其固有死亡（包括迁徙）率。

因此，该等式表示，捕食者种群的变化率取决于其捕杀猎物的速率减去其固有死亡（包括迁徙）率。

== Solutions to the equations 方程求解 ==

== Solutions to the equations 方程求解 ==